VIEMORANDUM

?.M-4577-PRTUNE 1965

SOME TABLES OFY'• THE NEGATIVE BINOMIAL DISTRIBUTION

AND THEIR USE

Bernice Brown

-- / - -

PREPARED FOR:

UNITED STATES AIR FORCE PROJEM;T RAND

_________Th7 r•i [1 DeCorpo7ateo

5ANTA MONICA CAtIFO'lrA------

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MEMORANDUM

RM-4577-PRJUNE 1965

SOME TABLES OFTHE NEGATIVE BINOMIAL DISTRIBUTION

AND THEIR USEBernice Brown

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_________________-- 74e~ 1Q4W I I I )eeAemauw

-iii-

In this Memorandum the author presents probability

tables of the negative binomial distribution for some

useful sets of parameter values. This distribution may

be used as a frame of reference for the study of the

demand for replacement parts.

M T~r -

SL,94ARY

This Memorandum presents tables giving the values of

the individual terms of the n,.gaivc 'Ainomial distribution

for 130 pairs of parameter values in Part I. Part 2,

giving the cumulated terms ,)f the same distributions, is

in a form directly usable for solving pjoblems which ze-

quire the determination of probabilities of the occurrence

of not more than x events.

The negativ% binomial .'],..ribution is described and

illustrativ., exat-t.ples are even which use this distribution

as a tool LI- Lhe study of demand for replacement parts.

The reterence- indicat -hat the negative binomial

has been us1 ,d in a wide vazL:,'•y of applications to bio-

logical research.

SOME. TABLES OF THE NEGATIVE BINOMIAL DISTRIBUTION

AND THEIR USE

L, INTRODUCTION

The research worker in the field of logistics "s

frequently required to predict the demand for spare parts

in order that effective decisions can be made in the areas

of procurement, distriuution, and stockage policies. Many

such decisions are made on the basis of a specialized

knowledge about an individual line item.

For routine decisionmaking on a large scale, it is

useful to have a small number of known probability distri-

butions which can be used as a framework to provide the

basic assumptions from which predicto.is of future demands

can be made. If demand data are available, an analyst

may study the sample data and make estimates of the param-

eters of the underlying demand distribution. The informa-

tion in the sample will often be scanty, e.g., six mornths

experience at one base or thrce months of system-wide

experience. A frame of reference is needed to study the

limited data.

Some examples of studies in which it was necessary to

make an assumption about the demand for spare parts in the

population may be found in the study of procurement defer-

ral by Petersen [Il, the design of flyaway kits by Karr,

Geisler, and Brown [2], and by Fort [3], the prediction of

-2-

demands by Goldman [4], and Geisler and Brown [51, the

study of stocka,;e policies by Ferguson [6], Ferguson and

Fisher [7], and Petersen and Geisler [8].

There are many distribution functions which could be

discussed. Only two are presented here, the Poisson and

the negative binomial. These two were chosen because they

are probability distributions of a discrete variable, and

demand for spare parts is by nature integral, and also

because we believe, hopefully, that they may describe the

universe of denmands from which the sample emanates. These

distribut'ons are attractive also because of their easy

computation.

2. THE POISSON DISTRIBUTION

When we deal with phenomena involving events that

occur randomly in time or particles that are randomly

distributed in space, the Poisson process is the model

used. An experiment is performed and "events" are tallied.

These events can be described by a function x = x(t), which

gives the number, x, of events observed during the first

t units of observation fur &ll values of t from 0 through

T (where T is the total number of observations). The re-

sults of such an experiment may be described by the Poisson

distribution if the events occur r in the sense of

the following definition. If any number of events x are

observed in any amount of time t, and if the points of the

occurrence of the x events are indapendently and uniformly

-3-

distributed between 0 and t, then the process may be de-

scribed a3 random. If the probability of the event is

small but a large number of independernt cases are taken,

the number of occurrences is likely to be distributed in

the Poisscn series. This distribution may be thought of

as an approximation to the binomial distribution when the

probability of occurrence of the event is small, that is,

if Np is large relative to p and N is large relative to

Np (where N is the number of trials and p is the probabili-

ty of the occurrence of the event in a single trial).

Let us try to visualize what the situation mig'at be

in regard to demand for spare parts upon the Air Force

Supply System. Suppose a mechanic inspects an actuator

on each of che 18 planes of a squadron, once each month

for two years, and requests a replacement part whenever he

judges that the actuator has failed. Assume that the fail-

ure rate for this part ha8 been found to be 25 in 1000 (i.e.,

1 in 40). We may then think of this situation as being

represented by a binomial distribution (p+q) 3 , where

N a 432 (18 inspections per month for 24 months), p a .025,

q a .975 (i.e., q a 1-V, the probability of nonoccurrence).

But the data available to us Is monthly data by squadron,

and we are interested in determining a squadron demand

rate per month. The monthly demand data for the actuator

looks like this: 0-2-O-0I-O--0-.l-O--l-----2-0---I-l--0-

0-1-0-1. These numbers are ordered in time, the first

observwtion being January, 1956 and the last one December,

1957.

-4-

This is a "natural' for the use of the Poisson approx-

imation to the binomial, since Np is large relative to p

(432 to 1) and N is large relative to Np (40 to 1). If

we assume that the Poisson law describes the data and use

as the parameter of the Poisson the mean demand per squadron

month, i.e., 18 x .025 - .45, we will find a satisfactory

fit with i0 ny.nths of no demand, 7 months of demand for

one part, and 2 months of demand for 2 parts.

The Poisson (like the binomial) is a distribution of

a discrete variable arising from enumeration data using

integral values only. Its basic characteristic is the

uniform probability of the occurrence of the event. In

contrast to both the binomial and the normal distribution,

it is defined by a single parameter, the mean. The variance

is equal to the mean. The Poisson density is represented

by the probability function P(x) a e-" mX/xI., where

x a 0,l,2,3'", and m a mean.

Tables of the Poisson distribution have been widely

published; for exaple see [9), 110), [111.

3. TI NAM TIYK Z UCIAAL MDZTBU'ION

But the aircraft demand data described in Sec. 2 does

not present this picture for many of the parts, and the

Poisson distribution often has not been a good fit for the

series of observations. In many cases, the lack of fit

was manifested in more vonths of zero demand thdn that

described by the Poisson distribution. It was also true

-5-

that we were getting more variation than was permitted

under the Poisson assumption. For these reansons and

others that will be discussed later, we used a distribution

known as the negative binomial to Ift the observed data.

It is a two-parameter distrihbtion of a discrete variable.

The follow, ng example illustrates this distribution.

The monthly deimand data fronm a squaeron of aircraft for a

door asse-mbly shows the following series of deinands over

a 36--month per 4 od: O-l-0--O---O-O-4-4-0-O.-, 0-0-0-1-0-0-0-

0-0-0-3-0-, 0-0-4-0-1-0-04---0-2-0-0. The sum of the de-

mands is 12. The mc-an demand per month is one-third. The

assumption of a Poisson distribution, using 1/3 as the

parameter Nialue, does not give a good fit to the data. The

negative !Anomial distribution, using the calculacion of

moments of tile obseived distribhtion to estimate che param-

eters. gives a good fit to the data. The ratio of variance

to neau• is *bout 2.•. If we uste an arbitrary ratio of

variance to mean of ',. the fit is also good. The inter-

pcetation of "good fit' means that the amount of discrepance

between the observed values and tt.e theoretical values

based .an the assumed distribution is not large enough to

.ndicate the presence of anything more than the capt.ces

of random sampling. In other words, the hypothesis is

upheld that this data could have been obtained from a

population in which monthly demand was described by a

negative binomial distr.bution.

-6-

The negative bioK.mial distribution is completely

defined by two parameters. the arithmetic mean m and a

positive exponent k. The distribution is written

(q - p)-k where p - n/k and p + 1 - q. The general term

in the expansion of this binomial gives the probability

P that an observation x will have values 0, 1, 2, "'.

The general term may be written

P(x) I _+_' x - 0,1,2,..., P, k > 0." (k--l)-"xI. q k+X ""

The curve defined by the value of P(x) is unimodal, so

that in fitting the negative binomial to an observed

distribution, any apparent bimodal or multimodal tendency

is attributed to random sampling. The negative binomial

is an extension of the Poisson series in which the popula-

tion mean m is not constant but varies contin, wusly in a

distribution which is proportional to that of Chi-squarr

(The distribution referred to is called Pearson Type III

or Gauma distribution). Thus the negative binomial may be

used to represent a composite of several Poisson distribu-

tions in which the number of observations per unit time

in repeated counts cannot be assumed to have the same

expected valut mean) in each unit of measurement. Student

[12] wrote in 1919 as follows: "If the presence of one

individual in a division increases the chance of other

individuals falling into that division, a negative binomial

will fit best, but if it decreases the chance, a positive

-7-

binomial." Bliss reported [13] on fitting the negative ,

binomnial to biological data. "The negative binomial is

thc easiest to compute and the most widely applicable of

the distributions for over-dispersion."

The negative binomial has been used in a variety of

applications to biological data. It was used by Bliss 13)

to fit a distribution function to counts of red mites on

apple leaves. It was used by Morgan et al [14] to describe

the distribution of bacterial clumps over a milk film.

Student [15] used it for a description of the counting of

yeast cells with a haemocyto.Teter. Stirritt [161 et al.

foud that it was adequate to describe the distribucion of

corn borers in a field experiment. Greenwood and Yule [17]

used a negative binomial to describe the dis~rib'rb on of

accidents experienced by machinists. (The theory was that

some machinists were more accident-prone than others.

There was aiso the possibility that shop conditions differed

from week to week in such a way as to cause accident risks

to vary during successive weeks of the period covered by

the data. In either case, the resulting distribution

mnight be expected to be of the fori of a negative binomial.)

In plant ecology, quadrant counts which deviated from the

Poisson distribution were attributed to the occurrence of

plants in "clumps." Blackman [18] found that the distri-

bl:tLon of plants per quadrant agreed very well with a

negative binomi.al distribution, with estimates of param-

ezers derived from the sample data. Jones, Mollison, and

-8-

Quenouille [19) used the negative binodal in fitting

counts of soil bacteria. Sichel (20] made use of a

negative binomial in studying psychological data on the

occurrence of minor accidents in an industrial plant.

Additional references could be cited, but our purpose

is merely to document the fact that the negative binomial

distribution has been used for years in widely divergent

fields of applicationz.

Here we are interested in the application of this

particular distribution to the demand for aircraft spare

parts. We shall first discuss the rationale which leads

us to believe that it may offer a reasonable description

of spare-parts demand.

Suppose that two-years' data on demand for spare

parts is available at a base. For illustrative purposes,

we will say that it is a bass with four squadrons, each

consisting of 18 aircraft. that is, the data covers the

demands of 72 planes for 24 months. If all the aircraft

were Identical, of the san ape, flew identical missions,

underwent identical servicito, -ere subject to the sawe

maintenance practices, etc., the demand for actuator parts

might be expected to be about th.e same as in the illustra-

tive example on page 4. In this case, the monthly demands

would follow the Poibson distribution. If the population

were homogeneous, random samples of demand data vould be

expected to exhibit only the sampling fluctuations in-.

herent in any ve1l-behaved variable. But the homogeneity

described above is not present in the Air Force environment

in which the demands are generated. For example, even

though the planes fly equal numbers of hours, make the same

number of landings, etc., some planes are nevertheless

subject to greater stresses than others because of factors

of speed, altitude, rate of climb, etc. The probability

of failure of a particular part is no longer constant, but

has increased due to the stre3s factor. The recorded

demands at the base for this part over the 24-month period

might be as follows: 1-0-7-0-0-3-1-2-0-0-6-I-, 0-1-0-4-1-1-

2-0-9-2-0-3. The sun, of the dcmuands at the base is 44.

There arc four squadrons, so that the demand rate per

squadron month is the same as in the Poisson-fitted example

on p. 4.

For these demand data, the Poisson is not a good fit.

If we use the X2 statistic to test the egreement of the

observed distribution with the theoretical Paisson distri-

bution, the computed X2 value for I d.f. is 8.3. The

probability of obtaining a X 2 value of this nasnit.ade or

greater in drawing random samples from a homogeneous

population is less than .01. Such large values of chi-

square may signify no more than the presence of an tun.-

usually divergent sample, but to the investigator who is

none too sure of his hypothesis, the presence of repeated

large chi-square values indicates that the hypothesis

should be rejected.

-10-

However, if we usc the negative binomial probability

distribution with the same mean demand per month ind assume

that the ratio of variance to mean is 3, we will obtain a

good fit for the sample data. (G.2 is .17 for 1 d.f.,

P = .70.) The discrepance between the observed frequency

distribution of demands per month and tho theoretical

frequencies based on the negative binomial is v'ery small

indeed. The ratio of sample variance to sample mean in

this case was about 3.2. A word of cauticn may be appro-

priate here concerning inferences made about the population

on the basis of the chi-square test of goodness of fit.

Th. viewpoint adopteO is that the hypothesis is fixed-

namely, that a negative binomial distribution describes

the population of monthly demand..s fur the part. The

probability evaluated above is that of drawing from the

population a sample more extreme than the one in hand. It

is not a method for evaluating the probability that the

hypothesis is correct. Of course, it is true that after

the evidence from samples is accumulate. the analyst-

researcher must make a decision about the hypothesis, and

his decision has some presumably high probability of being

correct; but we have not presented a method of evaluating

such a probability.

There are many factors in the Air Force environment

that ccntribute to heterogeneity of demands. Among these

factors we might list age of aircraft, applicability of

parts, flying-ipogram elements, nature of mission,

-11-

servicing schedules, maintenance practices, design change

and modifications, changes in base personnel, etc. A

crew mechanic inspecting plane number 1 may not have the

same careful standards as the mechanic who inspects number

2. Some inspectors are prone to replace parts-some are not.

Demands for certain spare parts have also been known to

exhibit a tendency to cccur in clusters--that is, the demand

for a unit of part a stimulates demand for a unit of part b,

which in turn creates a demand for 4 units of part c, etc.

All of these factors change with time, and are reflected in

the sample data.

4. TABLES OF NEGATIVE BINOKL DISTRIBUTION

We have found tables of the negative binomial distri-

bution useful in our work in logIstics. Since they may

not be readily accessible, the complete probability distri-

butions for a limLted number of arbitrary parameter values

are presented in the following paps. We have used 13

different values of the man (a a kp) and 10 different

ratios of varianc*-Co--man (q). This XLves us 130 sets

of parameter values for which the complete probability

distribution in tabulated.

The distribution function of the negative binomial is

P(x) - I

(k-)O. X0 q

where x = 001,203--, pok > 0, and q a 1 + p.

-12-

Part I gives the individual term of the distributions

and Part 2 gives the cumulative probability of x demands

co): less. The values of the mean (kp) are .25(.25)1.0 and

1.0(-,0)10.0. The values of the ratio of variance to mean

(q) zre 1.5(.5)5.0 and 5.0(1.0)7.0. The choice of these

values makes p vary from 1/2 to 6 and k from 1 to 20.

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-33-

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-34-

16. Stirrict, G. M., G. Beall, and M. Timonin, "A FieldExperiment on the Control of the European CornBoret," Sctentific Agriculture 17, 1937, pp. 587-591.

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